Summary: CONVOLUTION OPERATORS INDUCED BY APPROXIMATE
IDENTITIES AND POINTWISE CONVERGENCE IN Lp(R)
SPACES
PARTHENA AVRAMIDOU
Abstract. Given a sequence of kernels n for which the operators Tnf =
n f converge a.e. in all Lp(R) spaces, p 1, a perturbation method is
provided with the property that the modified convolution operators converge
pointwise only in selective spaces.
1. Introduction
Nagel and Stein [3] showed the potential of more general than nontangential
differentiation simultaneously in various Lp(Rn
) spaces. They obtained a necessary
and sufficient condition for the approach regions that determines boundedness of
the associated Hardy-Littlewood type maximal operator in all Lp(Rn
) spaces, p 1.
As an application, for kernels of the form n = 1
|In| In , where {In} is a sequence of
shrinking intervals approaching the origin, the corresponding convolution operators
either converge pointwise in all Lp(Rn
) spaces, p 1, or in none.